Stable Isomorphisms of Operator Algebras

Abstract

Abstract Let ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ be operator algebras with $c_{0}$-isomorphic diagonals and let ${{\mathcal{K}}}$ denote the compact operators. We show that if ${{\mathcal{A}}}\otimes{{\mathcal{K}}}$ and ${{\mathcal{B}}}\otimes{{\mathcal{K}}}$ are isometrically isomorphic, then ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ are isometrically isomorphic. If the algebras ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ satisfy an extra analyticity condition a similar result holds with ${{\mathcal{K}}}$ being replaced by any operator algebra containing the compact operators. For nonselfadjoint graph algebras this implies that the graph is a complete invariant for various types of isomorphisms, including stable isomorphisms, thus strengthening a recent result of Dor-On, Eilers, and Geffen. Similar results are proven for algebras whose diagonals satisfy cancellation and have $K_{0}$-groups isomorphic to ${{\mathbb{Z}}}$. This has implications in the study of stable isomorphisms between various semicrossed products.